Solve for $x$ and $y$ using elimination. $\begin{align*}x+2y &= 1 \\ -7x+6y &= -5\end{align*}$
Explanation: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-3$ and the bottom equation by $1$ $\begin{align*}-3x-6y &= -3\\ -7x+6y &= -5\end{align*}$ Add the top and bottom equations. $-10x = -8$ Divide both sides by $-10$ and reduce as necessary. $x = \dfrac{4}{5}$ Substitute $\dfrac{4}{5}$ for $x$ in the top equation. $ \dfrac{4}{5}+2y = 1$ $\dfrac{4}{5}+2y = 1$ $2y = \dfrac{1}{5}$ The solution is $\enspace x = \dfrac{4}{5}, \enspace y = \dfrac{1}{10}$.